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Theorem bm2.5ii 4222
Description: Problem 2.5(ii) of [BellMachover] p. 471. (Contributed by NM, 20-Sep-2003.)
Hypothesis
Ref Expression
bm2.5ii.1  |-  A  e. 
_V
Assertion
Ref Expression
bm2.5ii  |-  ( A 
C_  On  ->  U. A  =  |^| { x  e.  On  |  A. y  e.  A  y  C_  x } )
Distinct variable group:    x, y, A

Proof of Theorem bm2.5ii
StepHypRef Expression
1 bm2.5ii.1 . . 3  |-  A  e. 
_V
21ssonunii 4215 . 2  |-  ( A 
C_  On  ->  U. A  e.  On )
3 unissb 3610 . . . . . 6  |-  ( U. A  C_  x  <->  A. y  e.  A  y  C_  x )
43a1i 9 . . . . 5  |-  ( x  e.  On  ->  ( U. A  C_  x  <->  A. y  e.  A  y  C_  x ) )
54rabbiia 2547 . . . 4  |-  { x  e.  On  |  U. A  C_  x }  =  {
x  e.  On  |  A. y  e.  A  y  C_  x }
65inteqi 3619 . . 3  |-  |^| { x  e.  On  |  U. A  C_  x }  =  |^| { x  e.  On  |  A. y  e.  A  y  C_  x }
7 intmin 3635 . . 3  |-  ( U. A  e.  On  ->  |^|
{ x  e.  On  |  U. A  C_  x }  =  U. A )
86, 7syl5reqr 2087 . 2  |-  ( U. A  e.  On  ->  U. A  =  |^| { x  e.  On  |  A. y  e.  A  y  C_  x } )
92, 8syl 14 1  |-  ( A 
C_  On  ->  U. A  =  |^| { x  e.  On  |  A. y  e.  A  y  C_  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 98    = wceq 1243    e. wcel 1393   A.wral 2306   {crab 2310   _Vcvv 2557    C_ wss 2917   U.cuni 3580   |^|cint 3615   Oncon0 4100
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-un 4170
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-int 3616  df-tr 3855  df-iord 4103  df-on 4105
This theorem is referenced by: (None)
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